11/7/2018
1
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 1
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 30: Chap. 9 of F&W
Wave equation for sound in the linear approximation
1. Sound generation
2. Sound scattering
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 2
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 3
11/7/2018
2
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 4
01
2
2
22
tc
Solutions to wave equation:
22 where),(
:solution wavePlane
ckAet tii rkr
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 5
0 0
:onperturbatiin order lowest toEquations
0
0
vv
vfvv
v
tt
p
t
p
t applied
0 0
20 0
In terms of the velocity potential:
0
0 0
p p
t t
t t
v
v
v
Some comments about Monday’s lecture
0
0
Note that:
=
( )
0
, ,
for , , ' ( ')
t
pK t
t
t t
t t dt K t
p
t
r r
r r
v
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 6
2
00
0
),(
entropy (constant) thedenotes re whe),(
:density theof in terms pressure Expressing
cp
p
spp
sppspp
s
2
0 0
2 22
20 0
22 2 2
0 2
0 0
( ) 0
0 0
pc
t t
cc K t
t t t
ct t
Some comments about Monday’s lecture -- continued
11/7/2018
3
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 7
Some comments about Monday’s lecture -- continued
vs
pc
ct
2
222
2
Here,
0
:airfor equation Wave
0
Boundary values:
ˆImpenetrable surface with normal moving at velocity :
ˆ ˆ ˆ
Free surface:
0 0 Φ
pt
n V
n V n v n
2
22 2
2
22 2
2
0
0
Additional relations:
0
ct
c
t
pt
pc p
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 8
)'()'()','(1
where
)','()','(''),(
:function sGreen' of in termsSolution
),(1
2
2
22
3
2
2
22
ttttGtc
tfttGdtrdt
tftc
rrrr
rrrr
r
Wave equation with source:
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 9
'4
''
)','(
: thatshowcan We
rr
rr
rr
c
tt
ttG
Wave equation with source -- continued:
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4
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 10
Derivation of Green’s function for wave equation
dett
ttttGtc
tti '
2
2
22
2
1)'(
thatRecall
)'()'()','(1
rrrr
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 11
Derivation of Green’s function for wave equation -- continued
2
2222 where','
~
:satisfymust ,~
,~
2
1,
,,~
:Define
ckGk
G
deGtG
dtetGG
ti
ti
rrrr
r
rr
rr
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 12
Derivation of Green’s function for wave equation -- continued
0,~
,~1
,~
:0for and ' Define --Check
'4,'
~
:'in isotropy assumingSolution
','~
22
222
'
22
RGkRGRdR
d
RRGk
RR
eG
Gk
ik
rr
rrrr
rr
rrrr
rr
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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 13
Derivation of Green’s function for wave equation -- continued
R
eB
R
e ARG
BeA eRGR
RGRkRGRdR
d
RGkRGRdR
d
RRGk
R
ikRikR
ikRikR
,~
,~
0,~
,~
0,~
,~1
,~
:0For
22
2
22
222
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 14
Derivation of Green’s function for wave equation – continuedneed to find A and B.
4
1
''4
1 : thatNote 2
BA
rrrr
R
e RG
ikR
4,
~
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 15
Derivation of Green’s function for wave equation – continued
dee
dee
deGttG
ttii
ttiik
tti
c'
'
''
'
'42
1
'42
1
,'~
2
1','
rr
rr
rrrr
rr
rr
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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 16
Derivation of Green’s function for wave equation – continued
'4
''
','
2
1 that Noting
'42
1',' '
'
rr
rr
rr
rrrr
rr
ctt
ttG
u δ dω eπ
dee
ttG
ui
ttii c
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 17
In order to solve an inhomogenous wave equation with a time harmonic forcing term, we can use the corresponding Green’s function:
'4
,'~ '
rrrr
rr
ike G
In fact, this Green’s function is appropriate for solving equations with boundary conditions at infinity. For solving problems with surface boundary conditions where we know the boundary values or their gradients, the Green’s function must be modified.
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 18
SV
rdhgghrdhggh
)g()h(2322 ˆ: thatNote
and functions woConsider t
theoremsGreen'
n
rr
)'(),'(~
),(~~~
22
22
rrrr
r
Gk
fk
rdGG
rdfG
Ggh
V
2
S
3
ˆ),(~
,'~
,'~
),(~
,,'~
'),(~
~ ;
~
nrrrrrr
rrrrrr
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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 19
3
2
S
If the integration volume includes the point ':
( , ) ' , ', '
ˆ ( ', ) ' , ' , ( ', ) '
V
V
G f d r
G G d r
r r
r r r r
r r r r r r n
3
2
S
3
2
S
( , ) ' ' , ,
ˆ ( , ) ' , ' , ( , )
Exchanging ':
( ', ) ' ' , ', '
ˆ ( ', ) ' , ' , ( ', ) '
V
V
G f d r
G G d r
G f d r
G G d r
r r r r r r
r r r r r r n
r r
r r r r r r
r r r r r r n
extra contributions from boundary
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 20
),(1
2
2
22 tf
tcr
Wave equation with source:
Example:
ˆ( , ) time harmonic piston of radius , amplitude
can be represented as boundary value of ( , )
f t a
t
r z
r
z
yx
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 21
'ˆ),'(~
','~
,'~
'),'(~
','~
,'~
),(~
2
S
3
rdGG
rdfGV
nrrrrrr
rrrr
' '
Note: Need Green's function with vanishing gradient at 0 :
' , where ' '; 04 ' 4 '
ik ik
z
e eG z z z
r r r r
r rr r r r
Treatment of boundary values for time-harmonic force:
222
222
0for
for 0~
:exampleour for aluesBoundary v
ayxai
ayx
zz
11/7/2018
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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 22
''),'(
~,'
~),(
~
0'
dydxz
GS: z
rrrr
0 ;'2
,'~
0 ;'' re whe'4'4
,'~
0'
'
0'
''
ze
G
zzzee
G
z
ik
z
ikik
rrrr
rrrrrr
rr
rrrr
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 23
a
z
ik
S: z
eddrrai
dydxz
G
0
2
0 0'
'
0'
'2'''
''),'(
~,'
~),(
~
rr
rrrr
rr
'sinsin''ˆ'
ˆcosˆsinˆ
plane; yz in the is ˆ Assume
'ˆ' ;For
2
rrr
rar
rrrr
zyr
r
rrrr
'sin''
'cos'' :domainn Integratio
ry
rx
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 24
aikr
iu
aikr
ikr
krJdrrr
eai
uJed
eddrrr
eai
0
0
0
2
0
'sin
0
2
0
'sinsin'
)sin'(''),(~
)('2
1 : thatNote
'''2
),(~
r
r
sin
)sin(),(
~
)()(
13
0
10
ka
kaJ
r
eai
wwJuuduJ
ikr
w
r
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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 25
2
164320
2
*02
1
*21
sin
)sin(
2
1ˆ
:angle solidper power averaged Time
:average timeTaking
:fluxEnergy
ka
kaJakcr
dΩ
dP
i
p
p
e
e
e
rj
vj
vj
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 26
z
yx
2
164320
2
sin
)sin(
2
1ˆ
:angle solidper power averaged Time
ka
kaJakcr
dΩ
dPe rj
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 27
Figure from Fetter and Walecka pg. 337
Scattering of sound waves –for example, from a rigid cylinder
11/7/2018
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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 28
Scattering of sound waves –for example, from a rigid cylinder
( ) (
V
) ( ) ( )
elocity potential --
iinc sc inc e k rr r r r
2 2 22 2 2
2 22
2 2
Helmholz equation in cylindrical coordinates:
1 10
Assume:
1where ( ) 0
)
(im
m
m
m
k r kr rr r z
r
r
e R
d d mk R
dr r dr r
r r
r
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 29
cos( ) ( )i minc m
ikr im
m
e i e Je kr
k rr
( ) ( ) where Hankel function
represents an outgoing wave : ( ) ( ) ( )
Boundary condition at : 0
' ( )' ( ) ' ( ) 0
' ( )
imm m
m m m
r a
m m mm m
s
m m
cm
m
C kr
H kr J kr iN kr
r ar
J kai J ka C H ka C i
H k
H
a
e
r
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 30
' ( )( ) ( )
' ( )scm imm
mmm
J kai kr
H kH
ae
r
/ 42(
Asymptotic form:
) i krmm
kr
H kri ekr
/ 4
/4
' ( )1 2( )
' ( )
' ( )2
' ( )
sci krikr imm
kr m
i
m
m
mm
m
eJ ka
f er H ka kr
J kaf
k H ka
e
e
r
11/7/2018
11
11/7/2018 PHY 711 Fall 2018 -- Lecture 29 31
2df
d
For ka << 1
2 23 411 2cos
8k
df
da
/ 4' ( )2
' ( )m
i mm
m
J kaf
k H kae